BMO-type norms related to the perimeter of sets.

*(English)*Zbl 1352.46026The main question answered in the paper under review has its roots in [J. Bourgain et al., J. Eur. Math. Soc. (JEMS) 17, No. 9, 2083–2101 (2015; Zbl 1339.46028)]. This former article contains a proof that certain \(\mathbb{Z}\)-valued functions are constant. The proof is done by introducing a new function space. There the space of functions \(f\) is considered where the following quantity \([f]\), defined as follows, vanishes (integrals are principal values):

\[ \limsup_{\varepsilon\downarrow 0}\varepsilon^{n-1}\sup_{\mathcal{G}_\varepsilon} \sum_{Q'\in \mathcal{G}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx, \]

where \(\mathcal{G}_\varepsilon\) denotes a collection of disjoint \(\varepsilon\)-cubes \(Q'\subset Q\) with sides parallel to the coordinate axes and cardinality not exceeding \(\varepsilon^{1-n}\). In this earlier paper, it is proved that for measurable sets \(A\subset (0,1)^n\)

\[ \| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C[1_A]. \]

Denoting by \(P(A,(0,1)^n)\) the perimeter of \(A\) relative to \((0,1)^n\), one has the inequality

\[ \| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C\min\{1,P(A,(0,1)^n)\}. \]

In the paper under review, the authors look at an isotropic variant of \([f]\), since, after all, the perimeter is isotropic as well. They first define \[ I_\varepsilon(f):=\varepsilon^{n-1}\sup_{\mathcal{F}_\varepsilon}\sum_{Q'\in \mathcal{F}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx, \] where in opposition to \(\mathcal{G}_\varepsilon\), cubes in \(\mathcal{F}_\varepsilon\) are allowed to have arbitrary orientation.

The authors answer the question whether there is a relationship between \(I_\varepsilon(1_A)\) and \(P(A)\) by proving that for any measurable set \(A\subset \mathbb{R}^n\) \[ \lim_{\varepsilon\to 0} I_\varepsilon(1_A)=\frac{1}{2}\min\{1, P(A)\}. \] This equality also gives a sufficient condition for a measurable set \(A\) to have finite perimeter, namely \(\lim_{\varepsilon\to 0} I_\varepsilon(1_A)< 1/2\).

The proof of the main result takes several subsections. Establishing the upper bound \(I_\varepsilon(1_A)\) is only a few lines. The verification of the lower bound is split according to whether or not the set \(A\) has finite parameter. Finally, the last part of the proof addresses the one-dimensional setting.

In the last section, some variants are studied. Among them, a localized version of the main theorem is considered, and a new nondistributional characterization of the perimeter is given. The paper ends with a discussion of the relation of a variant of \([f]\) to the total variation of \(f\) and with an appendix which consists of the proof of an isoperimetric inequality first considered in [H. Hadwiger, Monatsh. Math. 76, 410–418 (1972; Zbl 0248.52012)] and later studied by other authors as well, yet not in the needed generality for the paper under review.

\[ \limsup_{\varepsilon\downarrow 0}\varepsilon^{n-1}\sup_{\mathcal{G}_\varepsilon} \sum_{Q'\in \mathcal{G}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx, \]

where \(\mathcal{G}_\varepsilon\) denotes a collection of disjoint \(\varepsilon\)-cubes \(Q'\subset Q\) with sides parallel to the coordinate axes and cardinality not exceeding \(\varepsilon^{1-n}\). In this earlier paper, it is proved that for measurable sets \(A\subset (0,1)^n\)

\[ \| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C[1_A]. \]

Denoting by \(P(A,(0,1)^n)\) the perimeter of \(A\) relative to \((0,1)^n\), one has the inequality

\[ \| 1_A-{\,-\hskip-1.08em\int}_{(0,1)^n} 1_A\|_{L^{n/(n-1)}((0,1)^n)}\leq C\min\{1,P(A,(0,1)^n)\}. \]

In the paper under review, the authors look at an isotropic variant of \([f]\), since, after all, the perimeter is isotropic as well. They first define \[ I_\varepsilon(f):=\varepsilon^{n-1}\sup_{\mathcal{F}_\varepsilon}\sum_{Q'\in \mathcal{F}_{\varepsilon}} {\,-\hskip-1.08em\int}_{Q'}| f(x)- {\,-\hskip-1.08em\int}_{Q'} f|\, dx, \] where in opposition to \(\mathcal{G}_\varepsilon\), cubes in \(\mathcal{F}_\varepsilon\) are allowed to have arbitrary orientation.

The authors answer the question whether there is a relationship between \(I_\varepsilon(1_A)\) and \(P(A)\) by proving that for any measurable set \(A\subset \mathbb{R}^n\) \[ \lim_{\varepsilon\to 0} I_\varepsilon(1_A)=\frac{1}{2}\min\{1, P(A)\}. \] This equality also gives a sufficient condition for a measurable set \(A\) to have finite perimeter, namely \(\lim_{\varepsilon\to 0} I_\varepsilon(1_A)< 1/2\).

The proof of the main result takes several subsections. Establishing the upper bound \(I_\varepsilon(1_A)\) is only a few lines. The verification of the lower bound is split according to whether or not the set \(A\) has finite parameter. Finally, the last part of the proof addresses the one-dimensional setting.

In the last section, some variants are studied. Among them, a localized version of the main theorem is considered, and a new nondistributional characterization of the perimeter is given. The paper ends with a discussion of the relation of a variant of \([f]\) to the total variation of \(f\) and with an appendix which consists of the proof of an isoperimetric inequality first considered in [H. Hadwiger, Monatsh. Math. 76, 410–418 (1972; Zbl 0248.52012)] and later studied by other authors as well, yet not in the needed generality for the paper under review.

Reviewer: Thomas Zürcher (Coventry)

##### MSC:

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |

42B35 | Function spaces arising in harmonic analysis |

28A75 | Length, area, volume, other geometric measure theory |

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\textit{L. Ambrosio} et al., Commun. Pure Appl. Math. 69, No. 6, 1062--1086 (2016; Zbl 1352.46026)

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